Prime ideal

Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of Z, i.e. there is a variety Xr defined over R such that Xr xr C = X. For every prime ideal p of R let Xp = XR xr R/p. There is a nonempty open subset U C pec(R) such that Xp is smooth for all p U, and the /-adic Betti-numbers of Xp coincide with those of X for all primes / different from the characteristic of A/p (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R/m is a finite field Fq of characteristic p /, we call Xm a good reduction of X modulo q. 

If the base ring k is not a field, then even maximal ideals turn out to be not quite all we want. The kernel of a homomorphism Z[X] - Z, for instance, is not maximal. The next natural generalization is to prime ideals, and these do indeed give a satisfactory theory. 

The spectrum Spec A of a ring A is the collection of its prime ideals. To see what topology it should have, consider fc. A closed set there is the set where a certain ideal I of functions vanishes the corresponding maximal ideals... 

Proof. Since nQA is reduced, it injects into it0(A/iy,.we must show the dimensions are the same. As / fc still consists of nilpotents, we may assume k = k. Then dim(7t0 A) is the number of connected components of Spec A. But since I is in every prime ideal, Spec A is homeomorphic to Spec(/ //). 

Lemma. Let A be an integral domain finitely generated over afield k. Let P be a nonzero prime ideal. The fraction field of A/P has lower transcendence degree than the fraction field of A. 

Domains of course have no nontrivial nilpotents, so a nilpotent element/ in R is in all prime ideals. Conversely, if/ is not nilpotent, take a maximal ideal J in Af its inverse image in R is prime and does.not containf Thus the set N of nilpotent elements in R is an ideal equal to the intersection of all prime ideals. One calls N the nilradical, and says R is reduced if N = 0. 

Going-up theorem of Cohen-Seidenberg. Let R be a ring (commutative as always) and S C R a subring such that R is integrally dependent on S. For all prime ideals P C S, there exist prime ideals P C R such that P fl S = P. 

Recall that by the noetherian decomposition theorem, if A C k [Xi,..., Xn is an ideal such that A = y/A, then A can be written in exactly one way as an intersection of a finite set of prime ideals, none of which contains any other. And a prime ideal is not the intersection of any two strictly bigger ideals. Therefore ... 

Proposition 2. In the bijection of the Corollary to Theorem 1, the irreducible algebraic sets correspond exactly to the prime ideals of k [X, ..., Xn]. Moreover, every closed algebraic set E can be written in exactly one way as ... 

Definition of Projective Varieties. Let P C k[Xo,..., Xn] be a homogeneous prime ideal, X = V(P) C Pn(k). We want to make X (with the Zariski topology) into a prevariety. We do it by defining a function field, getting local rings, and intersecting them, just as for affine varieties. 

Algebraic Version (Krull s Principal Ideal Theorem). Let R be a finitely generated integral domain over k, f R, P an isolated prime ideal of (/) (i.e., minimal among the prime ideals containing it). Then if f 0, tr.d-kR/P = tr.d. kR — 1-... 

In order to have an unambiguous notation, we shall write [P] for the element of Spec (R) given by the prime ideal P. This allows us to distinguish between the times when we think of P as an ideal in P, and the times when we think of [P] as a point in Spec (R). 

Spec (R) need not satisfy axiom Tl. In fact, the closure of [P] is exactly V(P), i.e., P1 P D P. Therefore [P] is a closed point if and only if P is a maximal ideal. At the other extreme, when R is an integral domain, (O) is a prime ideal and [(O)] is called the generic point of Spec (R) since its closure is the whole of Spec R. More generally, we define ... 

Proposition 1. Ifx E Spec (R), then the closure of x is irreducible andx is a generic point of this set. Conversely, every irreducible closed subset Z C Spec (R) equals V(P) for some prime ideal P C R, and [P] is its unique generic point. 

Proof. Spec (R) is the union of the Spec (R) fa s if and only if every point [P] does not contain some /<. This means that no prime ideal contains (..., fa,...), and this happens if and only if... 

Note that once we have Lemmas 1 and 2, the rest is purely sheaf-theoretic. One further point which is useful since our topological space has non-closed points, we have some maps between the stalks ox of ox. Suppose P C P2 are 2 prime ideals. Let Xi = [Pf]. Then x2 i, so every neighbourhood U of x2 contains xi this gives us a map ... 

Example B. Spec (fc[X]) the affine line over k. This is denoted A. k[X] has 2 types of prime ideals (o) and (f(X)), / an an irreducible polynomial. Therefore Spec (A [X]) has one closed point for each monic irreducible polynomial, and one generic point [(o)] whose closure is all of Spec (fc[X]). Assume k is algebraically closed. Then the closed points are all of the form [(X — o)] we call this the point X = a , and we find that A is just the ordinary X-line together with a generic point. The most general proper closed set is just a finite union of closed points. 

Example D. Almost identical comments apply to Spec (R) for any Dedekind domain R. In fact, all prime ideals are maximal or (0) hence again we have a line of closed points plus a generic point. (However, if R is not a P.I.D., we cannot conclude as before that all non-empty open sets are distinguished we know only that they are gotten by throwing out a finite set of closed points.)... 

Example E. = Spec (k[X,Y]), k algebraically closed. We get the maximal ideals (X — a, Y — b), the principal prime ideals (f(X, Y)), for / irreducible, and (0). By dimension theory there axe no other prime ideals. The set of maximal ideals gives us a set of closed points isomorphic to the usual X, F-plane. Then we must add one big generic point and for every irreducible curve, a point generic in that curve but not sticking out of it ... 

Example H. Spec (Z[X]). This is a so-called arithmetic surface and is the first example which has a real mixing of arithmetic and geometric properties. The prime ideals in Z[X] are ... 

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